- #1
LagrangeEuler
- 717
- 20
I currently styding applications of Lie groups and algebras in quantum mechanics.
[tex]U^{\dagger}(R)V_{\alpha}U(R)=\sum_{\beta}R_{\alpha \beta}V_{\beta} [/tex]
Where ##U(R)## represents rotation. Letter ##U## is used because it is unitary transformation and ##R_{\alpha \beta}## matrix elements of matrix of rotations. Why this is the way for vector transformation? Is there any explanation?
Also for me is interesting that
[tex]R_{\alpha \beta}=\delta_{\alpha \beta}+\omega_{\alpha \beta}[/tex]
And from that
[tex]U(R)=I+\frac{i}{2}\sum_{\mu \nu}\omega_{\mu \nu}J_{\mu \nu}[/tex]
[tex]U^{\dagger}(R)=I-\frac{i}{2}\sum_{\mu \nu}\omega_{\mu \nu}J_{\mu \nu}[/tex]
where ##\omega## is parameter and ##J## is generator of rotation. Second question. How to now what to take for ##U^{\dagger}(R)## and what for ##U(R)##? + or - sign.
[tex]U^{\dagger}(R)V_{\alpha}U(R)=\sum_{\beta}R_{\alpha \beta}V_{\beta} [/tex]
Where ##U(R)## represents rotation. Letter ##U## is used because it is unitary transformation and ##R_{\alpha \beta}## matrix elements of matrix of rotations. Why this is the way for vector transformation? Is there any explanation?
Also for me is interesting that
[tex]R_{\alpha \beta}=\delta_{\alpha \beta}+\omega_{\alpha \beta}[/tex]
And from that
[tex]U(R)=I+\frac{i}{2}\sum_{\mu \nu}\omega_{\mu \nu}J_{\mu \nu}[/tex]
[tex]U^{\dagger}(R)=I-\frac{i}{2}\sum_{\mu \nu}\omega_{\mu \nu}J_{\mu \nu}[/tex]
where ##\omega## is parameter and ##J## is generator of rotation. Second question. How to now what to take for ##U^{\dagger}(R)## and what for ##U(R)##? + or - sign.